A new class of $L_q$-norm zonoid depths

TL;DR

This paper introduces a new class of $L_q$-norm zonoid depths that overcome the outside problem of traditional zonoid depths, enhancing their applicability in multivariate analysis and classification.

Contribution

The paper proposes a novel $L_q$-norm zonoid depth that does not vanish outside the data convex hull, addressing a key limitation of existing depth functions.

Findings

The new depths have well-defined contours.

They solve the outside problem of traditional zonoid depths.

Examples illustrate their practical advantages.

Abstract

Zonoid depth, as a well-known ordering tool, has been widely used in multivariate analysis. However, since its depth value vanishes outside the convex hull of the data cloud, it suffers from the so-called `outside problem', which consequently hinders its many practical applications, e.g., classification. In this note, we propose a new class of \emph{$L_q$-norm zonoid depths}, which has no such problem. Examples are also provided to illustrate their contours.